Download Chapter_09_ParameterEstimation

Probability and Statistics
for Computer Scientists
Second Edition, By: Michael Baron
Section 9.1: Parameter
estimation
CIS 2033. Computational Probability and Statistics
Pei Wang
Parameters of distributions
After determining the family of distribution, the
next step is to estimate the parameters
Example 9.1: The number of defects on each
chip is believed to follow Pois(λ)
Since λ = E(X) is the expectation of a Poisson
variable, it can be estimated with a sample
mean X-bar
Method of moments
Method of moments (2)
Special cases: μ’2 is Var(X), and m’2 is s2(n-1)/n
Method of moments (3)
To estimate k parameters, we can equate the
first k population and sample moments (or
centralized version), i.e.
μ1 = m1 , … …, μk = mk
The left-hand sides of these equations depend
on the parameters, while the right-hand sides
can be computed from data
The method of moments estimator is the
solution of this system of equations
Moments method example
The CPU time for 30 randomly chosen tasks of
a certain type are (in seconds)
9 15 19 22 24 25 30 34 35 35
36 36 37 38 42 43 46 48 54 55
56 56 59 62 69 70 82 82 89 139
If they are considered to be the value of a
random variable X, what is the model?
Moments method example (2)
The histogram suggests Gamma distribution
Moments method example (3)
Moments method example (4)
From data, we compute
and write two equations,
Solving this system in terms of α and λ, we get
the method of moment estimates
Method of maximum likelihood
Maximum likelihood estimator is the
parameter value that maximizes the likelihood
of the observed sample, L(X1, …, Xn)
L(X1, …, Xn) is defined as p(X1, …, Xn) for a
discrete distribution, and f(X1, …, Xn) for a
continuous distribution
When the variables X1, …, Xn are independent,
L(X1, …, Xn) is obtained by multiplying the
marginal pmfs or pdfs
Maximum likelihood
Maximum likelihood estimator is the parameter
value that maximizes the likelihood L(θ) of the
observed sample, x1, …, xn
When the observations are independent of
each other, L(θ) =
pθ(x1)*...*pθ(xn) for a discrete variable
fθ(x1)*...*fθ(xn) for a continuous variable
Which is a function with θ as variable
Maximum likelihood (2)
Here we consider two types of L(θ):
1. If the function always increases or
decreases in its defined range, the
maximum value is at the boundary of
the range, i.e., the min or max of θ
2. If the function first increases then
decreases, the maximum value is at
where its derivative L’(θ) is zero
Example of Type 1
To estimate the θ in U(0, θ) given positive data
x1, …, xn, L(θ) is θ-n when θ ≥ max(x1, …, xn),
otherwise it is 0
So the best estimator for θ is max(x1, …, xn)
since L(θ) is a decreasing function when
θ ≥ max(x1, …, xn)
Similarly, if x1, …, xn are generated by U(a, b),
the maximum likelihood estimate is
a = min(x1, …, xn), b = max(x1, …, xn)
Example of Type 2
If the distribution is Ber(p), and m of the n
sample values are 1, L(p) = pm(1 – p)n–m
L’(p) = mpm–1(1 – p)n–m – pm(n – m)(1 – p)n–m–1
= (m – np)pm–1(1 – p)n–m–1
L’(p) is 0 when p = m/n, which also covers the
situation where p is 0 or 1
So the sample mean is a maximum likelihood
estimator of p in Ber(p)
Exercise
If a probability mass function is partially known,
how to guess the missing part using testing
data? Take the following die as an instance
a
1
p(a) 0.1
count 12
2
3
4
5
6
0.1
10
0.2
19
0.2
23
?
9
?
27
Log-likelihood
Log function turns multiplication into addition,
and power into multiplication
E.g. ln(f × g) = ln(f) + ln(g)
ln(fg) = g × ln(f)
Log-likelihood function and likelihood function
reach maximum at the same value
Therefore, ln(L(θ)) may be easier for getting
maximum likelihood
Log-likelihood (2)
E. g., L(p) = pm(1 – p)n–m
ln(L(p)) = m(ln(p)) + (n – m)(ln(1 – p))
[ln(L(p))]’ = m/p – (n – m)/(1 – p)
m/p – (n – m)/(1 – p) = 0
m/p = (n – m)/(1 – p)
m – mp = np – mp
p = m/n
Estimation of standard errors
Standard error equals Std(T), so can be
estimated through sample variances
Mean squared error
When both the bias and variance of estimators
can be obtained, usually we prefer the one that
has the smallest mean squared error (MSE)
For estimator T of parameter θ,
MSE(T) = E[(T − θ)2] = E[T2] −2θE[T] + θ2
= Var(T) + (E[T] − θ)2
= Var(T) + Bias(T)2
So, MSE summarizes variance and bias
MSE example
Let T1 and T2 be two unbiased estimators for
the same parameter θ based on a sample of
size n, and it is known that
Var(T1) = (θ + 1)(θ − n) / (3n)
Var(T2) = (θ + 1)(θ − n) / [(n + 2)n]
Since n + 2 > 3 when n > 1, MSE(T1) > MSE(T2) ,
so T2 is a better estimator for all values of θ
MSE example (2)
Let T1 and T2 be two estimators for the same
parameter, and it is known that
Var(T1) = 5/n2, Bias(T1) = -2/n
Var(T2) = 1/n2, Bias(T2) = 3/n
5 + 4 < 1 + 9, MSE(T1) < MSE(T2) , so T1 is a
better estimator for the parameter